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Homework Assignment 08: Central Potential Due: April 14, 2016 Problem 1 (28%) Consider the radial equation λ d2 R 2 dR + − 1 R = 0, + dr2 r dr r which corresponds to a bound-state problem (to potential of the A r2 + B r type). (1) [3%] Determine f (r), which is the asymptotic form of R(r), i.e., f (r) = R(r → ∞). (2) [5%] Write R(r) = f (r)h(r). Obtain the equation for h(r). (3) [8%] Write h(r) = rs (a0 + a1 r + · · · + aν rν + · · · ) and obtain the value of s. (Hint: Use boundary condition at r = 0 to check if these values make sense.) (4) [12%] Obtain the recursion relation for the coefficients above and, from it, the eigenvalues of λ. Note: You don’t need to find the eigenfunctions. Problem 2 (27%) In three dimensions, a spherically symmetric potential has the form of a delta-function away from the origin: V (r) = −λδ(r − a), where λ is a positive constant. (1) [18%] Find the eigenvalue equation for an s-wave bound state. (2) [9%] Show that λ has to have a minimum value for bound states to exist. What is that value? (Note: You might find this part is similar to Problem 2(b) in Homework 05.) Problem 3 (45%) This problem ask you to use spherical coordinates to solve the exact eigenvalues and eigenfunctions of the three-dimensional harmonic oscillator, 1 V (r) = mω 2 r2 . 2 following the instructions and notations below. (1) [20%] Set z = mωr2 /~, find the radial equation for u(z), where u(r) = rR(r) with R(r) the radial function. Find the asymptotic behaviors for u(z) at z → 0 and z → ∞, and show that u(z) can then be expressed as u(z) = z (l+1)/2 e−z/2 G(z). 1 (2) [15%] Find the equation for G(z), which should have the same format (but different parameters) as Eq. 4.43 in lecture notes for Coulomb potential. Then from recursion relation, obtain the eigenenergies. (3) [10%] Find the solutions for G(z), and then the eigenfunctions for R(r). (Note: You don’t need to evaluate the normalization parameter; but if you have time, you might use Mathematica to do so.) 2